On the eta invariant, stable positive scalar curvature, and Higher \(\hat A\)-genera (Q5948133)

Motivated by \textit{M. Gromov} and \textit{H. B. Lawson jun.} [Publ. Math., Inst. Hautes Etud. Sci. 58, 295-408 (1983; Zbl 0538.53047)] and \textit{J. Rosenberg} [ibid. 58, 409-424 (1983; Zbl 0526.53044)], the author finds some obstructions to odd-dimensional closed spin manifolds admitting metrics of positive scalar curvature. The first result is that if such an \(M\) admits a metric of positive scalar curvature, then the evaluation on \(M\)'s fundamental class of the cup product of \(\widehat{A} (M)\) with any product of degree-one cohomology classes must be zero. The essence of the proof is that the product of degree-one cohomology classes arises from a map to a product of circles, that the product of circles is the classifying space for a free abelian group, and that the strong Novikov conjecture is true for a free abelian group. The paper ends with a similar theorem in which the curvature hypothesis is that the manifold admits stably a metric of positive scalar curvature. This theorem includes an hypothesis about \(M\) mapping to a finite subcomplex of the classifying space of a group satisfying the strong Novikov conjecture. It appears to the reviewer that this hypothesis is used only to assert that the trivial vector bundle is the pullback of a bundle on such a classifying space. (Perhaps the reviewer is missing an important point.) As a tool in addressing whether \(M\) admits stably a metric of positive scalar curvature, the author proves some results that may be of independent interest. These results involve equality of eta invariants (or reduced eta invariants) of Dirac operators on sections of bundles (isomorphic vector bundles with different connections) over manifolds admitting (stably) metrics of positive scalar curvature.